Secondly, if a,b∈?ab?a,b\in\mathbb{E}, which should mean that line segments of lengths |a|a|a| and |b|b|b| are constructible, then we can easily construct line segments of lengths |a+b|ab|a+b| and |a-b|ab|a-b| by matching up endpoints of line segments.

If a,b∈?ab?a,b\in\mathbb{F}, then also a±bplus-or-minusaba\pm b, a⁢babab, and a/b∈?ab?a/b\in\mathbb{F}, the last of which is meaningful only when b≠0b0b\not=0;

3.

If z∈?∖{0}z?0z\in\mathbb{F}\setminus\{0\} and arg⁡(z)=θargzθ\operatorname{arg}(z)=\theta where 0≤θ<2⁢π0θ2π0\leq\theta<2\pi, then |z|⁢ei⁢θ2∈?zsuperscripteiθ2?\sqrt{|z|}e^{{\frac{i\theta}{2}}}\in\mathbb{F}.

In order to justify rule 2, all we need is the justification of rule 2 for ??\mathbb{E} along with the notion of copying an angle. For example, if a,b∈?ab?a,b\in\mathbb{F}, then the following picture can be made by copying an angle:

000bbba+baba+baaa

Finally to justify rule 3. If z∈?z?z\in\mathbb{F}, then |z|∈?z?|z|\in\mathbb{E}, so we have that |z|∈?z?\sqrt{|z|}\in\mathbb{E}. Since |z|⁢ei⁢θ=z∈?zsuperscripteiθz?|z|e^{{i\theta}}=z\in\mathbb{F}, we must have that an angle with measureθθ\theta is constructible. By the compass and straightedge construction of angle bisector, an angle with measureθ/2θ2\theta/2 is also constructible.